IMO 1959 solution of Q4

Statement: Construct a right angled triangle with given hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.


1st Solution:

Let |AB| = x and |AC| = x'

Then AD² = x.x'

Also x² + x'² = c² (since ABC is right angled at A).

Note that |AD| = c/2 since AD is the median of a right angled triangle with hypotenuse c units. (application of Apolonius Theorem)

Thus \(\frac{c^2}{4}=x\sqrt{c^2 - x^2}\) 

Hence we find 'x' in terms of c.

We draw a circle with diameter c. Cut off x = f(c) length from 'c' and construct the desired triangle.